Groups of permutations preserving orientation (parity) of subsets of a fixed size

We study permutations on n elements preserving orientation (parity) of every subset of size k. We describe all groups of these permutations. Unexpectedly, these groups (except some special cases) are either trivial or cyclic or dihedral. We produce these results while defining semigroups generalizing semigroups of order-preserving mappings and semigroups of orientation-preserving mappings.